Theorem zpnpcan2 | index | src |

theorem zpnpcan2 (a b c: nat): $ a +Z b -Z (a +Z c) = b -Z c $;
StepHypRefExpression
1 eqtr3
a +Z b -Z a -Z c = a +Z b -Z (a +Z c) -> a +Z b -Z a -Z c = b -Z c -> a +Z b -Z (a +Z c) = b -Z c
2 zsubsub
a +Z b -Z a -Z c = a +Z b -Z (a +Z c)
3 1, 2 ax_mp
a +Z b -Z a -Z c = b -Z c -> a +Z b -Z (a +Z c) = b -Z c
4 zsubeq1
a +Z b -Z a = b -> a +Z b -Z a -Z c = b -Z c
5 zpncan2
a +Z b -Z a = b
6 4, 5 ax_mp
a +Z b -Z a -Z c = b -Z c
7 3, 6 ax_mp
a +Z b -Z (a +Z c) = b -Z c

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp, itru), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_11, ax_12), axs_set (elab, ax_8), axs_the (theid, the0), axs_peano (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)